Simplify.jl

Simplify.jl implements methods for symbolic algebraic simplification in the Julia language.

Examples

Normalization involves determining the unique normal form of an expression ("simplest" equivalent expression) through repeated application of rules. Simplify.jl will use its internal set of algebraic rules by default, which includes trigonometry, logarithms, differentiation (based on DiffRules.jl), and more.

julia> @syms x y b θ;

julia> normalize(@term(1 / (sin(-θ) / cos(-θ))))
@term(1 / (-(sin(θ)) / cos(θ)))

julia> normalize(@term(log(b, 1 / (b^abs(x^2)))))
@term(log(b, 1 / b ^ abs(x ^ 2)))

julia> normalize(@term(diff(sin(2x) - log(x+y), x)))
@term(1 * -(inv(x + y) * (1 * diff(y, x) + 1 * one(x))) + 1 * cos(2x) * (2 * one(x) + x * 0))

julia> normalize(@term(!x & x | (y & (y | true))))
@term(!x & x | (y | true) & y)

julia> normalize(@term(y^(6 - 3log(x, x^2))))
@term(y ^ (-(6 * log(x, x)) + 6))

In many cases, it is useful to specify entirely custom rules by passing a Term Rewriting System as the second argument to normalize. This may be done either by manually constructing a Rules object or by using the RULES strategy for @term.

julia> @syms f g h;
       @vars x y;

julia> normalize(@term(f(x, f(y, y))), @term RULES [
          f(x, x) => 1
          f(x, 1) => x
       ])
@term(x)

julia> normalize(@term(f(g(f(1), h()))), Rules(
          @term(f(x)) => @term(x),
          @term(h())  => @term(3),
       ))
@term(g(1, 3))

julia> using Simplify: EvalRule

julia> normalize(@term(f(g(f(1), h()))), Rules(
          @term(f(x)) => @term(x),
          @term(h())  => @term(3),
          EvalRule(g, (a, b) -> 2a + b)
       ))
@term(5)

Variables may contain information about their domain, which may result in more specific normalizations.

julia> using SpecialSets

julia> @syms x y z;

julia> ctx = [get_context(); Image(y, GreaterThan(3)); Image(z, Even ∩ LessThan(0))];

julia> with_context(ctx) do
           normalize(@term(abs(x)))
       end
@term(abs(x))

julia> with_context(ctx) do
           normalize(@term(abs(y)))
       end
@term(y)

julia> with_context(ctx) do
           normalize(@term(abs(z)))
       end
@term(-z)

julia> ctx = [get_context(); Image(x, TypeSet(Int)); Image(y, TypeSet(Int))];

julia> with_context(ctx) do
           normalize(@term(diff(sin(2x) - log(x + y), x)))
       end
@term(cos(2x) * 2 + -(inv(x + y) * (diff(y, x) + 1)))

Acknowledgements

• Term Rewriting and All That
• @MasonProtter
• @Yijia-Chen